Actions of generalized derivations on prime ideals in $*$-rings with applications

Abstract

In this paper, we make use of generalized derivations to scrutinize the deportment of prime ideal satisfying certain algebraic $*$-identities in rings with involution. In specific cases, the structure of the quotient ring $\mathscr{R}/\mathscr{P}$ will be resolved, where $\mathscr{R}$ is an arbitrary ring and $\mathscr{P}$ is a prime ideal of $\mathscr{R}$ and we also find the behaviour of derivations associated with generalized derivations satisfying algebraic $*$-identities involving prime ideals. Finally, we conclude our paper with applications of the previous section's results.

Keywords

• Derivation
• generalized derivation
• involution
• prime ideal

References

1. [1] S. Ali and N.A. Dar, On $\ast$-centralizing mappings in rings with involution, Georgian Math. J. 21 (1), 25–28, 2014.
2. [2] S. Ali, N.A. Dar and A.N. Khan, On strong commutativity preserving like maps in rings with involution, Miskolc Math. Notes 16 (1), 17–24, 2015.
3. [3] K.I. Beidar, On functional identities and commuting additive mappings, Comm. Algebra 26, 1819–1850, 1998.
4. [4] K.I. Beidar,W.S. Martindale III and A.V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, New York: Marcel Dekker, Inc., 1996.
5. [5] H.E. Bell and M.N. Daif, On commutativity and strong commutativity-preserving maps, Canad. Math. Bull. 37 (4), 443–447, 1994.
6. [6] H.E. Bell and G. Mason, On derivations in near-rings and rings, Math. J. Okayama Univ. 34, 135-144, 1992.
7. [7] K. Bouchannafa, M.A. Idrissi and L. Oukhtite, Relationship between the structure of a quotient ring and the behavior of certain additive mappings, Commun. Korean Math. Soc. 37 (2), 359–370, 2022.
8. [8] K. Bouchannafa, A. Mamouni and L. Oukhtite, Structure of a quotient ring R/P and its relation with generalized derivations of R, Proyecciones Journal of Mathematics 41 (3), 623-642, 2022.

9. [9] M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1), 89–93, 1991.
10. [10] M. Brešar, W.S. Martindale III, C.R. Miers, Centralizing maps in prime rings with involution, J. Algebra 161, 342–357, 1993.
11. [11] N.A. Dar and A.N. Khan, Generalized derivations in rings with involution, Algebra Colloq. 24 (3), 393–399, 2017.
12. [12] Q. Deng and M. Ashraf, On strong commutativity preserving maps, Results Math. 30, 259–263, 1996.
13. [13] I.N. Herstein, Rings with involution, Chicago: The University of Chicago Press, 1976.
14. [14] B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (4), 1147–1166, 1998.
15. [15] M.A. Idrissi and L. Oukhtite, Structure of a quotient ring R/P with generalized derivations acting on the prime ideal P and some applications, Indian J. Pure Appl. Math. 53, 792–800, 2022.
16. [16] A.N. Khan and S. Ali, Involution on prime rings with endomorphisms, AIMS Math. 5 (4), 3274–3283, 2020.
17. [17] M.S. Khan, S. Ali and M. Ayed, Herstein’s theorem for prime idelas in rings with involution involving pairs of derivations, Comm. Algebra 50 (6), 2592–2603, 2022.
18. [18] C.K. Liu, Strong commutativity preserving generalized derivations on right ideals, Monatsh. Math. 166 (3-4), 453–465, 2012.
19. [19] C.K. Liu and P.K. Liau, Strong commutativity preserving generalized derivations on Lie ideals, Linear Multilinear Algebra 59 (8), 905–915, 2011.
20. [20] J. Ma, X.W. Xu and F.W. Niu, Strong commutativity-preserving generalized derivations on semiprime rings, Acta Math. Sin. (Engl. Ser.) 24 (11), 1835–1842, 2008.
21. [21] L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turk. J. Math. 38 (2), 225-232, 2014.
22. [22] M.A. Raza, A.N. Khan and H. Alhazmi, A characterization of b-generalized derivations on prime rings with involution, AIMS Math. 7 (2), 2413–2426, 2022.
23. [23] N. Rehman, M. Hongan and H.M. Alnoghashi, On generalized derivations involving prime ideals, Rend. Circ. Mat. Palermo Series 2 71, 601–609, 2022. https://doi.org/10.1007/s12215-021-00639-1.
24. [24] W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14, 29–35, 1976.